Microeconomic theory basic principles and extensions walter nicholson

www.elsolucionario.net CHAPTER THE MATHEMATICS OF OPTIMIZATION The problems in this chapter are primarily mathematical They are intended to give students some practice with taking derivatives and using the Lagrangian techniques, but the problems in themselves offer few economic insights Consequently, no commentary is provided All of the problems are relatively simple and instructors might choose from among them on the basis of how they wish to approach the teaching of the optimization methods in class Solutions U ( x, y )  x  y U U a = 8x , = 6y x y b 8, 12 U U c dU  dx + dy = x dx + y dy x y dy for dU  x dx  y dy  d dx dy 8 x 4 x = = dx y 3y e x  1, y  U  1    16 dy 4(1) f    2/3 dx 3(2) g U = 16 contour line is an ellipse centered at the origin With equation x  y  16 , slope of the line at (x, y) is 2.2 a www.elsolucionario.net 2.1 dy 4x  dx 3y Profits are given by   R  C  2q  40q  100 d *   4q  40 q  10 dq  *   2(10)  40(10)  100  100 d   so profits are maximized dq dR MR   70  2q dq b c www.elsolucionario.net  Solutions Manual dC  2q  30 dq so q* = 10 obeys MR = MC = 50 MC  2.3 Substitution: y   x so f  xy  x  x f   2x  x x = 0.5, y = 0.5, f = 0.25 £ = y  =0 x £ = x =0 y so, x = y using the constraint gives x  y  0.5, xy  0.25 2.4 Setting up the Lagrangian: ?  x  y   0.25  xy) £  1  y x £  1  x y So, x = y Using the constraint gives xy  x  0.25, x  y  0.5 2.5 a f (t )  0.5 gt  40t df 40   g t  40  0, t *  dt g b Substituting for t*, f (t * )  0.5 g (40 g )  40(40 g )  800 g c f (t * )  800 g g f   (t *) depends on g because t* depends on g g f 40 800   0.5(t * )2  0.5( )2  so g g g www.elsolucionario.net Note: f   2  This is a local and global maximum Lagrangian Method: ?  xy    x  y) www.elsolucionario.net Chapter 2/The Mathematics of Optimization  800 32  25, 800 32.1  24.92 , a reduction of 08 Notice that 800 g  800 322  0.8 so a 0.1 increase in g could be predicted to reduce height by 0.08 from the envelope theorem d 2.6 a This is the volume of a rectangular solid made from a piece of metal which is x by 3x with the defined corner squares removed V  x  16 xt  12t  Applying the quadratic formula to this expression yields b t 16 x  256 x  144 x 16 x  10.6 x   0.225 x, 1.11x To determine true 24 24  2V maximum must look at second derivative  16 x  24t which is negative only t for the first solution c If t  0.225 x, V  0.67 x3  04 x3  05x3  0.68x3 so V increases without limit d This would require a solution using the Lagrangian method The optimal solution requires solving three non-linear simultaneous equations—a task not undertaken here But it seems clear that the solution would involve a different relationship between t and x than in parts a-c 2.7 a Set up Lagrangian ?  x1  ln x2   (k  x1  x2 ) yields the first order conditions: £  1   x1 ?    x2 x2 £  k  x1  x2   Hence,    x2 or x2  With k = 10, optimal solution is x1  x2  b With k = 4, solving the first order conditions yields x2  5, x1  1 c Optimal solution is x1  0, x2  4, y  5ln Any positive value for x1 reduces y d If k = 20, optimal solution is x1  15, x2  Because x2 provides a diminishing marginal increment to y whereas x1 does not, all optimal solutions require that, once x2 reaches 5, any extra amounts be devoted entirely to x1 2.8 The proof is most easily accomplished through the use of the matrix algebra of quadratic forms See, for example, Mas Colell et al., pp 937–939 Intuitively, because concave functions lie below any tangent plane, their level curves must also be convex But the converse is not true Quasi-concave functions may exhibit ―increasing returns to scale‖; even though their level curves are convex, they may rise above the tangent plane when all variables are increased together www.elsolucionario.net t www.elsolucionario.net  Solutions Manual 2.9 a f1   x1  x 2  f   x1 x 2   f 22   (   1) x1 x 2 f12  f 21    x1 b c 1    1  x2 Clearly, all the terms in Equation 2.114 are negative If y  c  x1 x 2  /  1/  since α, β > 0, x2 is a convex function of x1 x  c x1 Using equation 2.98, f11 f 22  f 12   (  1) ( ) (  1) x12  x 22     x12  x 22 =   (1     ) x12 2.10 a b c 2 2  x2 2 which is negative for α + β > Since y  0, y  , the function is concave Because f11 , f 22  , and f12  f 21  , Equation 2.98 is satisfied and the function is concave   y is quasi-concave as is y But y is not concave for γ > All of these results can be shown by applying the various definitions to the partial derivatives of y www.elsolucionario.net f11   (  1) x1  x1  www.elsolucionario.net CHAPTER PREFERENCES AND UTILITY These problems provide some practice in examining utility functions by looking at indifference curve maps The primary focus is on illustrating the notion of a diminishing MRS in various contexts The concepts of the budget constraint and utility maximization are not used until the next chapter 3.1 This problem requires students to graph indifference curves for a variety of functions, some of which not exhibit a diminishing MRS 3.2 Introduces the formal definition of quasi-concavity (from Chapter 2) to be applied to the functions in Problem 3.1 3.3 This problem shows that diminishing marginal utility is not required to obtain a diminishing MRS All of the functions are monotonic transformations of one another, so this problem illustrates that diminishing MRS is preserved by monotonic transformations, but diminishing marginal utility is not 3.4 This problem focuses on whether some simple utility functions exhibit convex indifference curves 3.5 This problem is an exploration of the fixed-proportions utility function The problem also shows how such problems can be treated as a composite commodity 3.6 In this problem students are asked to provide a formal, utility-based explanation for a variety of advertising slogans The purpose is to get students to think mathematically about everyday expressions 3.7 This problem shows how initial endowments can be incorporated into utility theory 3.8 This problem offers a further exploration of the Cobb-Douglas function Part c provides an introduction to the linear expenditure system This application is treated in more detail in the Extensions to Chapter 3.9 This problem shows that independent marginal utilities illustrate one situation in which diminishing marginal utility ensures a diminishing MRS 3.10 This problem explores various features of the CES function with weighting on the two goods www.elsolucionario.net Comments on Problems www.elsolucionario.net  Solutions Manual Solutions 3.1 Here we calculate the MRS for each of these functions: a MRS  f x f y  — MRS is constant b MRS  f x fy  0.5( y x)0.5  y x — MRS is diminishing 0.5( y x) 0.5 c MRS  f x f y  0.5x0.5 — MRS is diminishing e MRS  f x f y  3.2 ( x  y ) y  xy ( x  y)2 ( x  y ) x  xy  y x — MRS is diminishing ( x  y)2 Because all of the first order partials are positive, we must only check the second order partials a f11  f 22  f  b f11 , f 22  0, f12  c f11  0, f 22  0, f12  Strictly quasiconcave Not strictly quasiconcave Strictly quasiconcave d Even if we only consider cases where x  y , both of the own second order partials are ambiguous and therefore the function is not necessarily strictly quasiconcave e 3.3 f11 , f 22  f12  Strictly quasiconcave a U x  y,U xx  0,U y  x,U yy  0, MRS  y x b U x  2xy2 ,U xx  y ,U y  2x2 y,U yy  2x2 , MRS  y x c U x  x ,U xx  1 x2 ,U y  y ,U yy  1 y , MRS  y x This shows that monotonic transformations may affect diminishing marginal utility, but not the MRS 3.4 a The case where the same good is limiting is uninteresting because U ( x1 , y1 )  x1  k  U ( x2 , y2 )  x2  U [( x1  x2 ) 2, ( y1  y2 ) 2]  ( x1  x2 ) If the limiting goods differ, then y1  x1  k  y2  x2 Hence, ( x1  x2 ) /  k and ( y1  y2 ) /  k so the indifference curve is convex www.elsolucionario.net d MRS  f x f y  0.5( x2  y )0.5  2x 0.5( x2  y )0.5  y  x y — MRS is increasing www.elsolucionario.net Chapter 3/Preference and Utility  b Again, the case where the same good is maximum is uninteresting If the goods differ, y1  x1  k  y2  x2 ( x1  x2 ) /  k , ( y1  y2 ) /  k so the indifference curve is concave, not convex 3.5 a U (h, b, m, r )  Min(h, 2b, m,0.5r ) b A fully condimented hot dog c $1.60 d $2.10 – an increase of 31 percent e Price would increase only to $1.725 – an increase of 7.8 percent f Raise prices so that a fully condimented hot dog rises in price to $2.60 This would be equivalent to a lump-sum reduction in purchasing power 3.6 a U ( p, b)  p  b b  2U  xcoke c U ( p, x)  U (1, x) for p > and all x d U (k , x)  U (d , x) for k = d e See the extensions to Chapter www.elsolucionario.net c Here ( x1  y1 )  k  ( x2  y2 )  [( x1  x2 ) / 2, ( y1  y2 ) / 2] so indifference curve is neither convex or concave – it is linear www.elsolucionario.net  Solutions Manual a b Any trading opportunities that differ from the MRS at x , y will provide the opportunity to raise utility (see figure) c A preference for the initial endowment will require that trading opportunities raise utility substantially This will be more likely if the trading opportunities and significantly different from the initial MRS (see figure) 3.8 a MRS   U / x  x 1 y     1  ( y / x) U / y  x y  This result does not depend on the sum α + β which, contrary to production theory, has no significance in consumer theory because utility is unique only up to a monotonic transformation b Mathematics follows directly from part a If α > β the individual values x relatively more highly; hence, dy dx  for x = y c The function is homothetic in ( x  x0 ) and ( y  y0 ) , but not in x and y 3.9 From problem 3.2, f12  implies diminishing MRS providing f11 , f 22  Conversely, the Cobb-Douglas has f12  0, f11 , f 22  , but also has a diminishing MRS (see problem 3.8a) 3.10 a MRS  U / x  x 1    ( y / x)1 so this function is homothetic U / y  y  1  b If δ = 1, MRS = α/β, a constant If δ = 0, MRS = α/β (y/x), which agrees with Problem 3.8 c For δ < 1 – δ > 0, so MRS diminishes d Follows from part a, if x = y MRS = α/β www.elsolucionario.net 3.7 www.elsolucionario.net Chapter 3/Preference and Utility  e With   5, MRS (.9)  MRS (1.1)    (.9)0.5  949     (1.1)0.5  1.05   With    1, MRS (.9)    (1.1)2  1.21   Hence, the MRS changes more dramatically when δ = –1 than when δ = 5; the lower δ is, the more sharply curved are the indifference curves When    , the indifference curves are L-shaped implying fixed proportions www.elsolucionario.net MRS (1.1)    (.9)2  81   www.elsolucionario.net Chapter 19/The Economics of Information  105 E(U) = ln(9,500) = 9.1590 Will buy insurance For brown without insurance E(U) = ln(9,000) + ln(10,000) = 9.1893 Better off without insurance c Since only blue buy insurance, fair premium is 800 Still pays this group to buy insurance [E(U) = 9.1269] Brown will still opt for no insurance E(U) = 9.1269 Brown premium = 200 E(U) = 9.1901 So Brown is better off under a policy that allows separate rate setting 19.5 a No separating equilibrium is possible since low-ability workers would always opt to purchase the educational signal identifying them as high-ability workers providing education costs less than $20,000 If education costs more than $20,000, no one would buy it b A high-ability worker would pay up to $20,000 for a diploma It must cost a lowability worker more than that to provide no incentive for him or her to buy it too 19.6 a U (18,000) = 9.7981 b U (18,300) = 9.8147 c Utility of Trip = 5U (18,200) + 5U (17,900) = 9.8009 So since expected utility from the trip exceeds the utility of buying from the known location, she will make the trip 19.7 a Here f ( p )  for 300  p  400 and f ( p ) = otherwise 100 p Cumulative function is F ( p)   300 x f ( x)dx  100 p | 300  p 3 100 For 300  p  400 F(p) = for P < 300 F(p) = for P > 400 105 www.elsolucionario.net d Blue premium = 800 www.elsolucionario.net Expected minimum price (see footnote of extension) is n p  300 400  n p   dp   300    dp 100   100  p   300  4   (n  1)  100   300  b n 1 400 |300 100 n 1 n p clearly diminishes with n: 2 n 3 2 d p / dn  200 (n  1)  n c Set dp nmin / dn  100(n  1)2   (n  1)  50 n = 6.07 (i.e., calls) An intuitive analysis is: With n = p nmin  316.67 With n = p nmin  314.29 With n = p nmin  312.50 So should stop at the 7th call 19.8 According to the Extensions, the searcher should choose p R so that C    0p R F ( p) dp   p  pR pR  p   300   dp  50   |300   100   100   p   50  R   so: p R  320  100  19.9 Patient utility maximization requires: U1c U 2c  pm Doctor Optimization requires: U1d pm  U 2d [U1c  pmU 2c ]  If U 2d  (which I interpret as meaning that the physician is a perfect altruist), this requires pm  U1c (U 2c  U1d ) Relative to patient maximization, 106 www.elsolucionario.net d p / dn  100 (n  1)  www.elsolucionario.net Chapter 19/The Economics of Information  107 this requires a smaller U 1c Hence, the doctor chooses more medical care than would a fully informed consumer 19.10 a Expected value of utility = 5(10) + 5(5) = 7.5 regardless of when coin is flipped b If coin is flipped before day 1, there is no uncertainty at day From the perspective of day 1, utility = 10 or with p = 0.5 so E1(U) = 5(10) + 5(5) = 7.5 If the coin is flipped at day two, E2(U) = 7.5 and E1[E2(U)]1 = 7.5 so date of flip does not matter c With α = 2, flipping at day yields 100 or 25 with p = 0.5 Flipping at day yields E2(U) = 5(10) + 5(5) = 7.5 and [E2(U)]2 = 56.25 < E1(U) Hence the individual prefers flipping at day d With α = 5, flipping at day yields utility of 10 or with p = 0.5 E1(U) = 2.70 Flipping at day yields E2(U) = 5(10) + 5(5) = 7.5 and [E2(U)].5 = 2.74 Hence, the individual prefers flipping at day e Utility is concave in c2, but expected utility is linear in utility outcomes if α = Timing doesn't matter With   , timing matters because utility values themselves are exponentiated with a day-1 flip, whereas expected utility values are exponentiated with a day-2 flip Values of α > favor a day-1 flip; values of α < favor a day-2 flip 107 www.elsolucionario.net E1(U) = 5(100) + 5(25) = 62.5 www.elsolucionario.net CHAPTER 20 The problems in this chapter illustrate how externalities in consumption or production can affect the optimal allocation of resources and, in some cases, describe the remedial action that may be appropriate Many of the problems have specific, numerical solutions, but a few (20.4 and 20.5) are essay-type questions that require extended discussion and, perhaps, some independent research Because the problems in the chapter are intended to be illustrative of the basic concepts introduced, many of the simpler ones may not full justice to the specific situation being described One particular conceptual shortcoming that characterizes most of the problems is that they not incorporate any behavioral theory of government—that is, they implicitly assume that governments will undertake the efficient solution (i.e., a Pigovian tax) when it is called for In discussion, students might be asked whether that is a reasonable assumption and how the theory might be modified to take actual government incentives into account Some of the material in Chapter 20 might serve as additional background to such a discussion Comments on Problems 20.1 An example of a Pigovian tax on output Instructors may wish to supplement this with a discussion of alternative ways to bring about the socially optimal reduction in output 20.2 A simple example of the externalities involved in the use of a common resource The allocational problem arises because average (rather than marginal) productivities are equated on the two lakes Although an optimal taxation approach is examined in the problem, students might be asked to investigate whether private ownership of Lake X would achieve the same result 20.3 Another example of externalities inherent in a common resource This question poses a nice introduction to discussing ―compulsory unitization‖ rules for oil fields and, more generally, for discussing issues in the market’s allocation of energy resources 20.4 This is a descriptive problem involving externalities, now in relation to product liability law For a fairly complete analysis of many of the legal issues posed here, see S Shavell, Economic Analysis of Accident Law 20.5 This is another discussion question that asks students to think about the relationship between various types of externalities and the choice of contract type The Cheung article on sharecropping listed in the Suggested Readings for Chapter 20 provides a useful analysis of some of the issues involved in this question 20.6 An illustration of the second-best principle to the externality issue Shows that the ability of a Pigovian tax to improve matters depends on the specific way in which the market is organized 108 www.elsolucionario.net EXTERNALITIES AND PUBLIC GOODS www.elsolucionario.net Chapter 20/Externalities and Public Goods  109 20.7 An algebraic public goods problem in which students are asked to sum demand curves vertically rather than horizontally 20.8 An algebraic example of the efficiency conditions that must hold when there is more than one public good in an economy 20.9 Another public goods problem In this case, the formulation is more general than in Problem 20.7 because there are assumed to be two goods and many (identical) individuals The problem is fairly easy if students begin by developing an expression for the RPT and for the MRS for each individual and then apply Equation 20.40 Solutions 20.1 a MC = 4q p = $20 Set p = MC 20 = 4q q = 50 b SMC = 5q Set p = SMC 20 = 5q q = 40 At the optimal production level of q = 40, the marginal cost of production is MC = 4q = 4(40) = 16, so the excise tax t = 20 –16 = $4 c 20.2 a F x  10l x - 0.5 l 2x F y  5l y First, show how total catch depends on the allocation of labor Lx + ly = 20 ly = 20 – lx T x y F =F +F T F  10l x  l x  (20  l x)  5l x  0.5l 2x  100 www.elsolucionario.net 20.10 This problem asks students to generalize the discussions of Nash and Lindahl equilibria in public goods demand to n individuals In general, inefficiencies are greater with n individuals than with only two www.elsolucionario.net 110  Solutions Manual Equating the average catch on each lake gives y x F  F :10  0.5  lx ly lx l x  10, l y  10 FT = 50 – 0.5(100) + 100 = 100 b max F T : 5l x  0.5 l 2x  100 T dF 5lx  dl x c Fxcase = 50 T l x  5, l y  15, F  112.5 average catch = 50/10 = x License fee on Lake X should be = 2.5 d The arrival of a new fisher on Lake X imposes an externality on the fishers already there in terms of a reduced average catch Lake X is treated as common property here If the lake were private property, its owner would choose LX to maximize the total catch less the opportunity cost of each fisher (the fish he/she can catch on Lake Y) So the problem is to maximize FX – 5lx which yields lx = as in the optimal allocation case 20.3 AC = MC = 1000/well a Produce where revenue/well = 1000 = 10q = 5000 –10n n = 400 There is an externality here because drilling another well reduces output in all wells b Produce where MVP = MC of well Total value: 5000n –10n2 MVP = 5000 –20n = 1000 n = 200 Let tax = x Want revenue/well –x = 1000 when n = 200 At n = 200, average revenue/well = 3000 So charge x = 2000 20.4 Under caveat emptor, buyers would assume all losses The demand curve under such a situation might be given by D Firms (which assume no liability) might have a horizontal long-run supply curve of S A change in liability assignment would shift both supply and demand curves Under caveat vendor, losses (of amount L) would now be incurred by firms, thereby shifting the long-run supply curve to S' www.elsolucionario.net F case = 37.5 average catch = 37.5/5 = 7.5 www.elsolucionario.net Individuals now no longer have to pay these losses and their demand curve will shift upward by L to D' In this example, then, market price rises from P1 to P2 (although the real cost of owning the good has not changed), and the level of production stays constant at Q* Only if there were major information costs associated with either the caveat emptor or caveat vendor positions might the two give different allocations It is also possible that L may be a function of liability assignment (the moral hazard problem), and this would also cause the equilibria to differ 20.5 There is considerable literature on this question, and a good answer should only be expected to indicate some of the more important issues Aspects of what might be mentioned include A specific services provided by landlords and tenants under the contracts B the risks inherent in various types of contracts, who bears these risks, and how is that likely to affect demand or supply decisions C costs of gathering information before the contract is concluded, and of enforcing the contract’s provisions D the incentives provided for tenant and landlord behavior under the contracts (for example, the incentives to make investments in new production techniques or to alter labor supply decisions) E ―noneconomic‖ aspects of the contracts such as components of landlords’ utility functions or historical property relationships www.elsolucionario.net Chapter 20/Externalities and Public Goods  111 www.elsolucionario.net 112  Solutions Manual In the diagram the untaxed monopoly produces QM at a price of PM If the marginal social cost is given by MC', QM is, in fact, the optimal production level A per-unit tax of t would cause the monopoly to produce output QR, which is below the optimal level Since a tax will always cause such an output restriction, the tax may improve matters only if the optimal output is less than QM, and even then, in many cases it will not 20.7 a To find the total demand for mosquito control, demand curves must be summed vertically Letting Q be the total quantity of mosquito control (which is equally consumed by the two individuals), the individuals’ marginal valuations are P = 100 – Q (for a) P = 200 – Q (for b) Hence, the total willingness to pay is given by 300 – 2Q Setting this equal to MC (= 120) yields optimal Q = 90 b In the private market, price will equal MC = 120 At this price (a) will demand 0, (b) will demand 80 Hence, output will be less than optimal c A tax price of 10 for (a) and 110 for (b) will result in each individual demanding Q = 90 and tax collections will exactly cover the per-unit cost of mosquito control 20.8 a For each public good (yi, i = 1, 2) the RPT of the good for the private good (a) should equal the sum of individuals' MRS's for the goods: n RPT  ( y i for x )   MRS ( y i for x) b For the two public goods (y1 and y2), the RPT between the goods should equal the ratio of the sums of the marginal utilities for each public good: RPT ( y1 for y  U )  MU i ( y1) i ( y 2) n n www.elsolucionario.net 20.6 www.elsolucionario.net Chapter 20/Externalities and Public Goods  113 20.9 a The solution here requires some assumption about how individuals form their expectations about what will be purchased by others If each assumes he or she can be a free rider, y will be zero as will be each person’s utility b Taking total differential of production possibility frontier 2x dx + 200y dy = gives dx 200 y 100 y   dy 2x x Individual MRS i  MU y  MU x 0.5 x i y 0.5  y x i x /100  y y xi For efficiency require sum of MRS should equal RPT x  MRS = y i i Hence, x 100 y  y x x = 10 y Using production possibility frontier yields 200y2 = 5000 y=5 x = 50 Utility = x/100 = 0.5 2.5 Ratio of per-unit tax share of y to the market price of x should be equal to the x MRS  i  y 10 n 20.10 a The condition for efficiency is that  MRS i  RPT The fact that the MRS’s are summed captures the assumption that each person consumes the same amount of the nonexclusive public good The fact that the RPT is independent of the level of consumers shows that the production of the good is nonrival b As in Equation 20.41, under a Nash equilibrium each person would opt for a share under which MRSi = RPT implies a much lower level of public good production than is efficient c Lindahl Equilibrium requires that i  MRSi / RPT and i  1.0 This would seem to pose even greater informational difficulties than in the two-person case www.elsolucionario.net RPT   www.elsolucionario.net www.elsolucionario.net 114 www.elsolucionario.net CHAPTER 21 POLITICAL ECONOMICS The problems in this final chapter are of two general types First are four problems in traditional welfare economics (Problems 21.1–21.3 and 21.5) that illustrate various issues that arise in comparing utility among individuals These are rather similar to the problems in Chapter 12 The other six problems in the chapter concern public choice theory 21.1 A problem utilizing two very simple utility functions to show how none of several differing welfare criteria seems necessarily superior to all the others This clearly illustrates the basic dilemma of traditional welfare economics 21.2 This problem examines the Scitovsky bribe criterion for judging welfare improvements Although the criterion as a general principle is not widely accepted, the notion of “bribes” in public policy discussions is still quite prevalent (for example, in connection with trade adjustment policies) 21.3 Shows how to integrate production into the utility possibility frontier construction In the example given here, the frontiers are concentric ellipses so the Pareto criterion suggests choosing the one that is furthest from the origin The choice is, however, ambiguous if the frontiers intersect 21.4 Illustrates the “irrelevant alternative” assumption in the Arrow theorem 21.5 A further examination of welfare criteria that focuses on Rawls’ uncertainty issues Shows that the results derived from a Rawls’ “initial position” depend crucially on the strategies individuals adopt in risky situations 21.6 Further examination of the Arrow theorem and of how contradictions can arise in fairly simple situations 21.7 A simple problem focusing on an individual’s choice for the parameters of an unemployment insurance policy The problem would need to be generalized to provide testable implications about voting (see the Persson and Tabellini reference) 21.8 A problem in rent seeking The main point is to differentiate between the allocational harm of monopoly itself and the transfer nature of rent-seeking expenditures 21.9 A discussion question concerning voter participation 114 www.elsolucionario.net Comments on Problems www.elsolucionario.net Chapter 21/Political Economics  115 21.10 An alternative specification for probabilistic voting that also yields desirable normative consequences Solutions U1= 200 pounds a 100 pounds each b f1 f1 = 40 c U  U  f2 U1 = 10, U2 = f 1 f2 U2= f1 f www.elsolucionario.net 21.1 f2 = 160 f1 200  f 0.5 0.5 f  (200  f 1) f1 = 160, f2 = 40 d U  , best choice is U2 =  f2 = 100, f1 = 100 技 技 0.5 0.5 W     f f f (200  f ) U U 2 e 2 W ? 1 3/  /4  f [  (200  f ) ]  f (200  f )   f1 2 f (200  f ) ? f1 = 200 – f1 21.2  /4 3/  f1 (200  f ) f1 = 100, f2 = 100 If compensation is not actually made, the bribe criterion amounts to assuming that total dollars and total utility are commensurable across individuals As an example, consider: Income in State A Income in State B Individual 100,000 110,000 Individual 5,000 State B is “superior” to State A in that Individual could bribe Individual But, in the absence of compensation actually being made, it is hard to argue that State B is better 115 www.elsolucionario.net 21.3 Pareto efficiency requires MRS1 = MRS2 y1  x1 y2  x2 y  y1 x  x1 Hence, all efficient allocations have y1   y x1   x y  (1   ) y x  (1   ) x U  (1   ) xy 2 U   xy (U  U )  U 12  2U U  U 22   xy  2( )(1   ) xy  (1   ) xy  xy a If y  10, x  160 Utility Frontier is (U1 + U2)2 = 1600 b y  30 x  120 (U1 + U2)2 = 3600 c Maximize XY subject to X  2Y  180 yields x  90 y  45 (U1 + U2)2 = 4050 d In this problem, the utility possibility frontiers not intersect, so there is no ambiguity in using the Pareto criterion If they did intersect, however, one would want to use an outer envelope of the frontiers 21.4 individuals with states A, B, C Votes are A B C 2 If C is not available, let both C votes go to B A B This example is quite reasonable: it implies that Arrow’s axiom is rather restrictive 21.5 a D b E, E(U) = 5(30) + 5(84) = 57 c E(U) = 6(L) + 4(H) EUA = 50, EUB = 52, EUC = 48.6, EUD = 51.5, EUE = 50 So choose B 116 www.elsolucionario.net www.elsolucionario.net Chapter 21/Political Economics  117 d max E(U) – | U  U | values: A: 50 – = 50 C: 49.5 – = 40.5 E: 57 – 54 = B: 55 – 30 = 26 D: 51.75 – 2.5 = 49.26 So choose A e It shows that a variety of different choices might be made depending on the criteria being used 21.6 Suppose preferences are as follows: Preference C A B A B C B C A a Under majority rule, APB (where P means “is socially preferred to”), BPC, but CPA Hence, the transivity axiom is violated b Suppose Individual is very averse to A and reaches an agreement with Individual to vote for C over B if Individual will vote for B over A Now, majority rule results in CPA, CPB, and BPA The final preference violates the nondictatorship assumption since B is preferred to A only by Individual c With point voting, each option would get six votes, so AIBIC But that result can be easily overturned by introducing an “irrelevant alternative” (D) 21.7 a So long as this utility function exhibits diminishing marginal utility of income, this person will opt for parameters that yield y1 = y2 Here that requires w(1 – t) = b Inserting this into the governmental budget constraint produces uw(1 – t) = tw(1 – u) which requires u = t b A change in u will change the tax rate by an identical amount c The solutions in parts a and b are independent of the risk aversion parameter,  21.8 a Since p = –q/100 + 2, MR = –q/50 + MR = MC when q = 75, p = 1.25, π = 56.25 The firm would be willing to pay up to this amount to obtain the concession (assuming that competitive results would otherwise obtain) b The bribes are a transfer, not a welfare cost c The welfare loss is the deadweight loss from monopolization of this market, which here amounts to 28.125 117 www.elsolucionario.net Individual www.elsolucionario.net 21.9 An essay on this topic would stress that free riding may be a major problem in elections where voters perceive that the marginal gain from voting may be quite small If such voters are systematically different from other voters, candidates will recognize this fact and tailor their platforms to those who vote rather than to the entire electorate The effect would be ameliorated by the extent to which platforms can affect voter participation itself 21.10 Candidate 1’s problem is to chose θ1, to maximize n i 1 i 1   i   f i (U i ( 1i ) / U i ( 2i ) subject to n  1i  i 1 The first order conditions for a maximum are fi ' U i' / U i ( *2i )   for all i  n Assuming f i ' is the same for all individuals, this yields U i' / U i ( *2i )  k  for i  n In words, the candidate should equate the ratio of the marginal utilities of any two voters (Ui' / U 'j ) to the ratio of their total utilities (U i / U j ) Since each candidate follows this strategy, they will adopt the strategies that would maximize the Nash Function, SWF 118 www.elsolucionario.net n ... 5/Income and Substitution Effects  21 sandwich In part a, for example, a ten percent increase in the price of ham will increase the price of a sandwich by percent and that will cause quantity demanded... e Since David N uses only pb + j to make sandwiches (in fixed proportions), and because bread is free, it is just as though he buys sandwiches where psandwich = 2ppb + pj www.elsolucionario.net... c d Increases in I shifts demand for x outward Reductions in py not affect demand for x until p y < px Then demand for x falls to zero e The income-compensated demand curve for good x is the